Heavy Tail Behavior and Parameters Estimation of GARCH (1, 1) Process

In practice, Financial Time Series have serious volatility cluster, that is large volatility tend to be concentrated in a certain period of time, and small volatility tend to be concentrated in another period of time. While GARCH models can well describe the dynamic changes of the volatility of financial time series, and capture the cluster and heteroscedasticity phenomena. At the beginning of this paper, the definitions and basic theories of GARCH(1,1) models are discussed. Secondly, show the heavy tail behavior of GARCH(1,1) process with α -stable residuals { } t t Z ε ∈ , (0, 2] α ∈ and { } t t Z ε ∈ errors. In fact, both these processes have heavy-tailed properties, but generally the tail of GARCH(1,1) process is heavier than the tail of { } t t Z ε ∈ errors. And then the modification of maximum likelihood function has been constructed as the theoretical basis of this study, make use of Holder inequality and Jensen's inequality to estimate parameters of GARCH(1,1) model with residuals having regularly varying distributions with index 0 α > . Finally, the consistency and asymptotic normality of the estimates constructed are further proved.